a: \(A=2+2^2+\cdots+2^{2010}\)
\(=\left(2+2^2\right)+\left(2^3+2^4\right)+\cdots+\left(2^{2009}+2^{2010}\right)\)
\(=2\left(1+2\right)+2^3\left(1+2\right)+\cdots+2^{2009}\left(1+2\right)\)
\(=3\left(2+2^3+\cdots+2^{2009}\right)\) ⋮3
Ta có: \(A=2+2^2+\cdots+2^{2010}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+\cdots+\left(2^{2008}+2^{2009}+2^{2010}\right)\)
\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+\cdots+2^{2008}\left(1+2+2^2\right)\)
\(=7\left(2+2^4+\cdots+2^{2008}\right)\) ⋮7
b: Ta có: \(B=3+3^2+3^3+\cdots+3^{2010}\)
\(=\left(3+3^2\right)+\left(3^3+3^4\right)+\cdots+\left(3^{2009}+3^{2010}\right)\)
\(=3\left(1+3\right)+3^3\left(1+3\right)+\cdots+3^{2009}\left(1+3\right)\)
\(=4\left(3+3^3+\cdots+3^{2009}\right)\) ⋮4
Ta có: \(B=3+3^2+3^3+\cdots+3^{2010}\)
\(=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+\cdots+\left(3^{2008}+3^{2009}+3^{2010}\right)\)
\(=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+\cdots+3^{2008}\left(1+3+3^2\right)\)
\(=13\left(3+3^4+\cdots+3^{2008}\right)\) ⋮13
c: \(C=5+5^2+\cdots+5^{2010}\)
\(=\left(5+5^2\right)+\left(5^3+5^4\right)+\cdots+\left(5^{2009}+5^{2010}\right)\)
\(=5\left(1+5\right)+5^3\left(1+5\right)+\cdots+5^{2009}\left(1+5\right)\)
\(=6\left(5+5^3+\cdots+5^{2009}\right)\) ⋮6
Ta có: \(C=5+5^2+\cdots+5^{2010}\)
\(=\left(5+5^2+5^3\right)+\left(5^4+5^5+5^6\right)+\cdots+\left(5^{2008}+5^{2009}+5^{2010}\right)\)
\(=5\left(1+5+5^2\right)+5^4\left(1+5+5^2\right)+\cdots+5^{2008}\left(1+5+5^2\right)\)
\(=31\left(5+5^4+\cdots+5^{2008}\right)\)
=>C⋮31
